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            <h1 style="display: none">algorithm-扫描线</h1>
            
              <p class="note note-info">
                
                  本文最后更新于：7 个月前
                
              </p>
            
            <div class="markdown-body" id="post-body">
              <h2 id="0-x-前言"><a href="#0-x-前言" class="headerlink" title="[0.x]前言"></a>[0.x]前言</h2><p>扫描线算法(<code>ScanLine</code>)用于求矩阵面积并或周长并，内核为线段树维护离散化的线段并在线处理。</p>
<p>语言：$Cpp$</p>
<p>为了节省时间，图片全部引自<a target="_blank" rel="noopener" href="https://ncc79601.blog.luogu.org/scan-line">这篇Luogu题解</a></p>
<a id="more"></a>
<h2 id="目录"><a href="#目录" class="headerlink" title="目录"></a>目录</h2><ul>
<li>[1.x]扫描线概述</li>
<li>[2.x]扫描线实现<ul>
<li>[2.1.x]预处理&amp;离散化</li>
<li>[2.2.x]线段树维护</li>
<li>[2.3.x]在线处理</li>
</ul>
</li>
<li>[3.x]相关代码</li>
</ul>
<hr>
<h2 id="1-x-扫描线概述"><a href="#1-x-扫描线概述" class="headerlink" title="[1.x]扫描线概述"></a>[1.x]扫描线概述</h2><p>扫描线一般用于处理以下问题：</p>
<p>读入若干矩阵，求矩阵的周长并/面积并。</p>
<p><img src="https://img.ffis.me/images/2019/08/10/image.png" srcset="/blog/img/loading.gif" style="zoom:25%;" /></p>
<p>该问题的处理难点在于：</p>
<ul>
<li>当用点来维护矩阵的时候，数值可能过大，需要离散化</li>
<li>维护完点集以后，暴力计算处理繁复而且缓慢，需要线段树维护</li>
</ul>
<p>我们用线段树维护从下到上的每一个线段，每一次向上推移一个离散后的单位长度，进行求解，就可以得到每一部分的面积</p>
<p><img src="https://img.ffis.me/images/2019/08/10/image65ecdba6bd78c3ce.png" srcset="/blog/img/loading.gif" style="zoom:50%;" /></p>
<ul>
<li>如我们维护<code>x[1]</code>~<code>x[3]</code>的长度，乘以两条红色线之间的距离，就是这个绿色区域的面积</li>
</ul>
<h2 id="2-x-扫描线实现"><a href="#2-x-扫描线实现" class="headerlink" title="[2.x]扫描线实现"></a>[2.x]扫描线实现</h2><h3 id="2-1-x-预处理-amp-离散化"><a href="#2-1-x-预处理-amp-离散化" class="headerlink" title="[2.1.x]预处理&amp;离散化"></a>[2.1.x]预处理&amp;离散化</h3><p>我们需要预处理的东西主要有以下几个：</p>
<ul>
<li>维护<strong>每一条横向线段</strong>，包括<strong>左端点</strong>和<strong>右端点</strong>，以及<strong>线段的高度</strong>和属性（即这条边是一个矩形的<strong>上边还是下边</strong>）</li>
</ul>
<div class="hljs"><pre><code class="hljs cpp">line[++tot].l = x_1 , line[tot].r = x_2 , line[tot].h = y_1 , line[tot].v = <span class="hljs-number">1</span>;
line[++tot].l = x_1 , line[tot].r = x_2 , line[tot].h = y_2 , line[tot].v = <span class="hljs-number">-1</span>;
<span class="hljs-comment">//我们定义下边的权值为+1，因为我们扫到一个下边就意味着有一个新的区域要加入计算</span>
<span class="hljs-comment">//我们定义上边的权值为-1，因为我们扫到一个上边就意味着有一个旧的区域要脱离计算</span>
<span class="hljs-built_in">std</span>::sort(line+<span class="hljs-number">1</span>,line+tot_line+<span class="hljs-number">1</span>,cmp);
<span class="hljs-comment">//按照高度(line[tot].h)排序，因为我们处理的时候是从下往上处理的</span></code></pre></div>
<ul>
<li>维护所有端点的<strong>横座标</strong>并去重以便<strong>离散化</strong></li>
</ul>
<div class="hljs"><pre><code class="hljs cpp"><span class="hljs-comment">//两个代码框之间的tot不共通，只是为了缩小代码块码量，请主要区分</span>
point[++tot] = x_1 , point[++tot] = x_2;
<span class="hljs-comment">//存入所有的横座标</span>
<span class="hljs-built_in">std</span>::sort(point+<span class="hljs-number">1</span>,point+tot_point+<span class="hljs-number">1</span>,cmp);
tot_x = <span class="hljs-built_in">std</span>::unique(point+<span class="hljs-number">1</span>,point+tot+<span class="hljs-number">1</span>)-point<span class="hljs-number">-1</span>;
<span class="hljs-comment">//排序后去重，用来离散化</span></code></pre></div>
<h3 id="2-2-x-线段树维护"><a href="#2-2-x-线段树维护" class="headerlink" title="[2.2.x]线段树维护"></a>[2.2.x]线段树维护</h3><p>我们用线段树维护线段而非点。</p>
<p>对于一个节点<code>cur</code>，我们有这么几个属性：</p>
<ul>
<li><code>cur-&gt;l</code>，表示离散化后的该线段左端点序号</li>
<li><code>cur-&gt;r</code>，表示离散化后的该线段右端点序号的前一个，即真正的右端点应该为<code>point[cur-&gt;r+1]</code></li>
<li><code>cur-&gt;len</code>，表示当前节点所表示的线段里已经被标记过的有多长</li>
<li><code>cur-&gt;sum</code>，表示该线段被标记与否<ul>
<li>其实这里可以用<code>bool</code>类型，但是我们刚才定义了上下边为<code>1/-1</code>，所以可以用<code>int</code>的加减来实现<code>0/1</code>即<code>T/F</code>的转换</li>
</ul>
</li>
<li><code>cur-&gt;ls</code> <code>cur-&gt;rs</code>，左儿子右儿子，不解释</li>
</ul>
<div class="hljs"><pre><code class="hljs cpp"><span class="hljs-class"><span class="hljs-keyword">struct</span> <span class="hljs-title">node</span>&#123;</span>
    <span class="hljs-keyword">int</span> l,r;
    ll sum,len;
    node * ls, * rs;
&#125;Tree[<span class="hljs-number">4</span>*N];</code></pre></div>
<p>其中特别要注意的就是，我们维护的是线段，而<code>cur-&gt;r</code>这个属性维护的前一个序号，这就能保证线段树的线段没有重复但是维护的线段端点可以重合。</p>
<p>另外，之所以这里不用打<code>tag</code>，是因为这里有个特殊的性质：</p>
<ul>
<li>对于一个节点，也就是对于一条特定的线段，它一定存在偶数个待处理线段，这是由输入的东西是矩阵保证的。换句话说，你要是这个节点被标记了，那说明这上面的一定都可以，不存在因为子节点的修改而导致该节点的变化。所以既不需要<code>pushdown</code>也不需要<code>lazy_tag</code>。</li>
<li>再换句话说，对于线段上的任意一点，只要经过改点的线段<strong>有一条</strong>是被标记的，就可以纳入计算。</li>
</ul>
<div class="hljs"><pre><code class="hljs cpp"><span class="hljs-function"><span class="hljs-keyword">inline</span> node * <span class="hljs-title">create</span><span class="hljs-params">()</span></span>&#123;<span class="hljs-keyword">return</span> &amp;Tree[++tot_tree];&#125; <span class="hljs-comment">// 线段树申请节点</span>
<span class="hljs-function"><span class="hljs-keyword">inline</span> <span class="hljs-keyword">void</span> <span class="hljs-title">build</span><span class="hljs-params">(node * cur,<span class="hljs-keyword">int</span> l,<span class="hljs-keyword">int</span> r)</span></span>&#123; <span class="hljs-comment">// 线段树建树</span>
    cur-&gt;l = l , cur-&gt;r = r , cur-&gt;sum = cur-&gt;len = <span class="hljs-number">0</span>; <span class="hljs-comment">// 节点初始化</span>
    <span class="hljs-keyword">if</span>(l == r) <span class="hljs-keyword">return</span>;
    <span class="hljs-keyword">int</span> mid = (l+r)&gt;&gt;<span class="hljs-number">1</span>;
    cur-&gt;ls = create() , cur-&gt;rs = create();
    build(cur-&gt;ls,l,mid) , build(cur-&gt;rs,mid+<span class="hljs-number">1</span>,r);
    <span class="hljs-keyword">return</span>;
&#125;
<span class="hljs-function"><span class="hljs-keyword">inline</span> <span class="hljs-keyword">void</span> <span class="hljs-title">pushup</span><span class="hljs-params">(node * cur)</span></span>&#123; <span class="hljs-comment">// 喜闻乐见的pushup</span>
    <span class="hljs-keyword">if</span>(cur-&gt;sum) cur-&gt;len = point[cur-&gt;r+<span class="hljs-number">1</span>] - point[cur-&gt;l]; <span class="hljs-comment">// 如果当前节点被标记就直接用当前线段长度</span>
    <span class="hljs-keyword">else</span> cur-&gt;len = cur-&gt;ls-&gt;len + cur-&gt;rs-&gt;len; <span class="hljs-comment">// 当前节点没被标记就用儿子的</span>
&#125;
<span class="hljs-function"><span class="hljs-keyword">inline</span> <span class="hljs-keyword">void</span> <span class="hljs-title">edit</span><span class="hljs-params">(node * cur,<span class="hljs-keyword">int</span> k)</span></span>&#123;
    <span class="hljs-keyword">int</span> l = cur-&gt;l , r = cur-&gt;r;
    <span class="hljs-keyword">if</span>(L &gt;= point[r+<span class="hljs-number">1</span>] || R &lt;= point[l]) <span class="hljs-keyword">return</span>;
    <span class="hljs-keyword">if</span>(L &lt;= point[l] &amp;&amp; R &gt;= point[r+<span class="hljs-number">1</span>])&#123;
        cur-&gt;sum += k;
        <span class="hljs-keyword">if</span>(cur-&gt;l != cur-&gt;r) pushup(cur);
        <span class="hljs-keyword">else</span>&#123; <span class="hljs-keyword">if</span>(cur-&gt;sum) cur-&gt;len = point[cur-&gt;r+<span class="hljs-number">1</span>] - point[cur-&gt;l];<span class="hljs-keyword">else</span> cur-&gt;len = <span class="hljs-number">0</span>;&#125;
        <span class="hljs-keyword">return</span>;
    &#125;
    edit(cur-&gt;ls,k) , edit(cur-&gt;rs,k);
    pushup(cur);
    <span class="hljs-keyword">return</span>;
&#125;
<span class="hljs-function"><span class="hljs-keyword">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span></span>&#123;
    ...
    node * root = create();
    build(root,<span class="hljs-number">1</span>,tot_x<span class="hljs-number">-1</span>); <span class="hljs-comment">// n个点，n-1个独立线段</span>
    ...
&#125;</code></pre></div>
<h3 id="2-3-x-在线处理"><a href="#2-3-x-在线处理" class="headerlink" title="[2.3.x]在线处理"></a>[2.3.x]在线处理</h3><p>我们现在已经用线段树维护这些线段了，也维护了从下到上的线段，我们只需要从下到上一个个枚举这些线段就行了。</p>
<div class="hljs"><pre><code class="hljs cpp">rep(i,<span class="hljs-number">1</span>,tot_line)&#123;
    L = line[i].l , R = line[i].r; <span class="hljs-comment">// 全局变量减少函数传参</span>
    edit(root,line[i].v);pushup(root); <span class="hljs-comment">// 在线维护扫描线上的标记长度</span>
    ans += root-&gt;len * (line[i+<span class="hljs-number">1</span>].h - line[i].h); <span class="hljs-comment">// 计算</span>
&#125;</code></pre></div>
<h2 id="3-x-相关代码"><a href="#3-x-相关代码" class="headerlink" title="[3.x]相关代码"></a>[3.x]相关代码</h2><p><a href="P5490 [模板]扫描线">P5490 【模板】扫描线</a></p>
<div class='spoiler collapsed'>
    <div class='spoiler-title'>
        Code
    </div>
    <div class='spoiler-content'>
        <div class="hljs"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;map&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;set&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;cmath&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;queue&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;bitset&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;vector&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;cstdio&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;cstring&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;cstdlib&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;iostream&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;algorithm&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">define</span> rep(i,a,b) for(register int i = (a);i &lt;= (b);++i)</span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">define</span> per(i,a,b) for(register int i = (a);i &gt;= (b);--i)  </span>
<span class="hljs-keyword">typedef</span> <span class="hljs-keyword">long</span> <span class="hljs-keyword">long</span> ll;
<span class="hljs-keyword">typedef</span> <span class="hljs-keyword">unsigned</span> <span class="hljs-keyword">long</span> <span class="hljs-keyword">long</span> ull;
<span class="hljs-keyword">using</span> <span class="hljs-built_in">std</span>::<span class="hljs-built_in">string</span>;<span class="hljs-keyword">using</span> <span class="hljs-built_in">std</span>::<span class="hljs-built_in">cin</span>;<span class="hljs-keyword">using</span> <span class="hljs-built_in">std</span>::<span class="hljs-built_in">cout</span>;

<span class="hljs-keyword">const</span> <span class="hljs-keyword">int</span> N = <span class="hljs-number">100010</span>;
<span class="hljs-keyword">int</span> n,x_1,x_2,y_1,y_2,tot_line,tot_x,tot_point,tot_tree,L,R;
ll point[<span class="hljs-number">4</span>*N],ans;

<span class="hljs-class"><span class="hljs-keyword">struct</span> <span class="hljs-title">LINE</span>&#123;</span>
    <span class="hljs-keyword">int</span> l,r,h,v;
&#125;line[<span class="hljs-number">4</span>*N];

<span class="hljs-class"><span class="hljs-keyword">struct</span> <span class="hljs-title">node</span>&#123;</span>
    <span class="hljs-keyword">int</span> l,r;
    ll sum,len;
    node * ls, * rs;
&#125;Tree[<span class="hljs-number">4</span>*N];

<span class="hljs-function"><span class="hljs-keyword">inline</span> <span class="hljs-keyword">bool</span> <span class="hljs-title">cmp1</span><span class="hljs-params">(LINE a,LINE b)</span></span>&#123;<span class="hljs-keyword">return</span> a.h&lt;b.h;&#125;
<span class="hljs-function"><span class="hljs-keyword">inline</span> <span class="hljs-keyword">bool</span> <span class="hljs-title">cmp2</span><span class="hljs-params">(<span class="hljs-keyword">int</span> x,<span class="hljs-keyword">int</span> y)</span></span>&#123;<span class="hljs-keyword">return</span> x&lt;y;&#125;

<span class="hljs-function"><span class="hljs-keyword">inline</span> node * <span class="hljs-title">create</span><span class="hljs-params">()</span></span>&#123;<span class="hljs-keyword">return</span> &amp;Tree[++tot_tree];&#125;

<span class="hljs-function"><span class="hljs-keyword">inline</span> <span class="hljs-keyword">void</span> <span class="hljs-title">build</span><span class="hljs-params">(node * cur,<span class="hljs-keyword">int</span> l,<span class="hljs-keyword">int</span> r)</span></span>&#123;
    cur-&gt;l = l , cur-&gt;r = r , cur-&gt;sum = cur-&gt;len = <span class="hljs-number">0</span>;
    <span class="hljs-keyword">if</span>(l == r) <span class="hljs-keyword">return</span>;
    <span class="hljs-keyword">int</span> mid = (l+r)&gt;&gt;<span class="hljs-number">1</span>;
    cur-&gt;ls = create() , cur-&gt;rs = create();
    build(cur-&gt;ls,l,mid) , build(cur-&gt;rs,mid+<span class="hljs-number">1</span>,r);
    <span class="hljs-keyword">return</span>;
&#125;

<span class="hljs-function"><span class="hljs-keyword">inline</span> <span class="hljs-keyword">void</span> <span class="hljs-title">pushup</span><span class="hljs-params">(node * cur)</span></span>&#123;
    <span class="hljs-keyword">if</span>(cur-&gt;sum) cur-&gt;len = point[cur-&gt;r+<span class="hljs-number">1</span>] - point[cur-&gt;l];
    <span class="hljs-keyword">else</span> cur-&gt;len = cur-&gt;ls-&gt;len + cur-&gt;rs-&gt;len;
&#125;

<span class="hljs-function"><span class="hljs-keyword">inline</span> <span class="hljs-keyword">void</span> <span class="hljs-title">edit</span><span class="hljs-params">(node * cur,<span class="hljs-keyword">int</span> k)</span></span>&#123;
    <span class="hljs-keyword">int</span> l = cur-&gt;l , r = cur-&gt;r;
    <span class="hljs-keyword">if</span>(L &gt;= point[r+<span class="hljs-number">1</span>] || R &lt;= point[l]) <span class="hljs-keyword">return</span>;
    <span class="hljs-keyword">if</span>(L &lt;= point[l] &amp;&amp; R &gt;= point[r+<span class="hljs-number">1</span>])&#123;
        cur-&gt;sum += k;
        <span class="hljs-keyword">if</span>(cur-&gt;l != cur-&gt;r) pushup(cur);
        <span class="hljs-keyword">else</span>&#123; <span class="hljs-keyword">if</span>(cur-&gt;sum) cur-&gt;len = point[cur-&gt;r+<span class="hljs-number">1</span>] - point[cur-&gt;l];<span class="hljs-keyword">else</span> cur-&gt;len = <span class="hljs-number">0</span>;&#125;
        <span class="hljs-keyword">return</span>;
    &#125;
    edit(cur-&gt;ls,k) , edit(cur-&gt;rs,k);
    pushup(cur);
    <span class="hljs-keyword">return</span>;
&#125;

<span class="hljs-function"><span class="hljs-keyword">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span></span>&#123;
    <span class="hljs-built_in">std</span>::ios::sync_with_stdio(<span class="hljs-number">0</span>);<span class="hljs-built_in">cin</span>.tie(<span class="hljs-number">0</span>);<span class="hljs-built_in">cout</span>.tie(<span class="hljs-number">0</span>);
    <span class="hljs-comment">//freopen(&quot;in.in&quot;, &quot;r&quot;, stdin);</span>
    <span class="hljs-built_in">cin</span> &gt;&gt; n;
    rep(i,<span class="hljs-number">1</span>,n)&#123;
        <span class="hljs-built_in">cin</span> &gt;&gt; x_1 &gt;&gt; y_1 &gt;&gt; x_2 &gt;&gt; y_2;
        line[++tot_line].l = x_1 , line[tot_line].r = x_2 , line[tot_line].h = y_1 , line[tot_line].v = <span class="hljs-number">1</span>;
        line[++tot_line].l = x_1 , line[tot_line].r = x_2 , line[tot_line].h = y_2 , line[tot_line].v = <span class="hljs-number">-1</span>;
        point[++tot_point] = x_1 , point[++tot_point] = x_2; 
    &#125;
    <span class="hljs-built_in">std</span>::sort(line+<span class="hljs-number">1</span>,line+tot_line+<span class="hljs-number">1</span>,cmp1) , <span class="hljs-built_in">std</span>::sort(point+<span class="hljs-number">1</span>,point+tot_point+<span class="hljs-number">1</span>,cmp2);
    tot_x = <span class="hljs-built_in">std</span>::unique(point+<span class="hljs-number">1</span>,point+tot_point+<span class="hljs-number">1</span>)-point<span class="hljs-number">-1</span>;
    node * root = create();
    build(root,<span class="hljs-number">1</span>,tot_x<span class="hljs-number">-1</span>);
    rep(i,<span class="hljs-number">1</span>,tot_line)&#123;
        L = line[i].l , R = line[i].r;
        edit(root,line[i].v);pushup(root);
        ans += root-&gt;len * (line[i+<span class="hljs-number">1</span>].h - line[i].h);
    &#125;
    <span class="hljs-built_in">cout</span> &lt;&lt; ans &lt;&lt; <span class="hljs-string">&quot;\n&quot;</span>;
    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;
&#125;</code></pre></div>

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